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"Tensor dapat dipandang sebagai multidimensional array, yang merupakan perluasan konsep dari matriks, sehingga dapat dikonstruksi bentuk umum dari hasil kali tensor yang sifatnya hampir sama dengan sifat perkalian pada matriks. Pada skripsi ini disajikan teori-teori yang digunakan seperti polinomial homogen, monomial berderajat alpha, dan resultan 2 polinomial untuk menjadi landasan teori utama dalam memahami determinan tensor. Tujuan dari skripsi ini adalah memperlihatkan dan memahami sifat-sifat determinan tensor.

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The tensor can be viewed as a multidimensional array, which is an extension of the concept of the matrix, so that the general form of the tensor result can be constructed which is almost the same as the multiplication property of the matrix. This undergraduate thesis presents the theories used such as homogeneous polynomials, monomial degrees of alpha, and resultant 2 polynomials to form the basis of the main theory in understanding tensor determinants. The purpose of this paper is to show the properties of tensor determinants."

"Tensor yang dipandang sebagai multidimensional array adalah bentuk umum dari suatu matriks. Oleh karena itu, dapat dikonstruksi bentuk umum dari hasil kali matriks yang disebut sebagai hasil kali tensor. Tujuan dari tulisan ini adalah menjelaskan inversi kiri dan inversi kanan suatu tensor. Pada tulisan ini disajikan karakteristik eksistensi inversi kiri dan inversi kanan orde k dari suatu tensor. Disajikan pula hasil terkait keserupaan suatu tensor.

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Tensor, which is seemed as multidimensional array, is a general form of matrix. Therefore, tensor could be constructed into general form of matrix product which is called tensor product. The aim of this writing was to explain the right and left inversion of tensor. In this research, there were characteristics of right and left extension of orde k of tensor provided, in addition, there was also a result involved of the tensor similarity."

"Tensor yang dipandang sebagai multidimensional array adalah bentuk umum dari suatu matriks. Oleh karena itu, dapat dikonstruksi bentuk umum dari hasil kali matriks yang disebut sebagai hasil kali tensor. Tujuan dari tulisan ini adalah menjelaskan inversi kiri dan inversi kanan suatu tensor. Pada tulisan ini disajikan karakteristik eksistensi inversi kiri dan inversi kanan orde k dari suatu tensor. Disajikan pula hasil terkait keserupaan suatu tensor

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Tensor, which is seemed as multidimensional array, is a general form of matrix. Therefore, tensor could be constructed into general form of matrix product which is called tensor product. The aim of this writing was to explain the right and left inversion of tensor. In this research, there were characteristics of right and left extension of orde k of tensor provided, in addition, there was also a result involved of the tensor similarity.

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"Dalam konteks matematika komputasi, tensor sering dipandang sebagai larik multidimensi, dengan jumlah dimensinya disebut sebagai orde tensor tersebut. Tensor dapat digunakan untuk merepresentasikan berbagai jenis data, seperti data gambar dan data psikometri. Salah satu masalah yang penting dalam komputasi tensor adalah aproksimasi rank rendah tensor. Untuk sebuah tensor A, masalah aproksimasi rank rendah adalah mencari tensor B yang nilainya paling mendekati tensor A tetapi memiliki rank tertentu yang lebih kecil dari rank A. Untuk tensor orde 2 (matriks), Teorema Eckart-Young-Mirsky menjelaskan bahwa masalah aproksimasi rank rendah matriks dapat diselesaikan dengan dekomposisi nilai singular (SVD). Akan tetapi, memperumum Teorema Eckart-Young-Mirsky untuk tensor adalah sebuah persoalan yang rumit. Masalah utamanya adalah, dalam kasus tensor, ada beberapa definisi rank yang berbeda. Masing-masing definisi rank dihasilkan dengan memperumum sifat-sifat tertentu dari fungsi rank matriks dan dapat menghasilkan nilai yang berbeda-beda untuk tensor yang sama; permasalahan tersebut adalah pokok bahasan skripsi ini. Skripsi ini dimulai dengan membahas konsep-konsep dasar dalam komputasi tensor. Lalu, akan dibahas mengenai tiga definisi konsep rank tensor. Untuk masing-masing definisi rank tensor, akan dipaparkan dekomposisi tensor yang berkaitan; dekomposisi-dekomposisi tensor ditujukan untuk memperumum SVD. Lalu, konsep rank dan dekomposisi tensor digabungkan dalam pembahasan masalah aproksimasi rank tensor. Pembahasan dilanjutkan dengan pembahasan hasil kali *M. Hasil kali *M dibuat untuk membentuk sebuah kerangka umum sebagai upaya menggabungkan beberapa dekomposisi tensor yang telah dibahas sebelumnya. Terakhir, dijelaskan mengenai berbagai sifat dan keunggulan teoretis kerangka hasil kali *M.

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In the context of computational mathematics, tensors are often viewed as multidimensional arrays, with the number of dimensions referred to as the order of the tensor. Tensors can be used to represent various types of data, such as image data and psychometric data. One important problem in tensor computation is the low-rank approximation of tensors. For a tensor A, the low-rank approximation problem is to find the tensor B whose entries are closest to the tensor A but has a certain rank that is smaller than the rank of A. For tensors of order two (matrices), the Eckart-Young-Mirsky theorem says that the matrix low-rank approximation problem can be solved by truncating its singular value decomposition (SVD). However, generalizing the Eckart-Young-Mirsky theorem to tensors is a complicated problem. The main problem is that there are several different definitions of rank in the case of tensors. Each definition of rank is generated by generalizing certain properties of the matrix rank and can yield different values for the same tensor; that problem is the subject of this thesis. This thesis begins by discussing the basic concepts of tensor computation. Then, three definitions of the concept of rank tensor will be addressed. For each definition of rank tensor, the corresponding tensor decomposition is presented; the tensor decompositions are intended to generalize the SVD. Then, the concepts of rank and tensor decomposition are combined to discuss the rank tensor approximation problem. The discussion continues with the discussion of the product of *M. The product of *M is made to form a general framework as an attempt to combine several tensor decompositions that have been discussed previously. Finally, various properties and theoretical advantages of the *M product framework are explained."

"Permeabilitas batuan merupakan parameter penting dalam meningkatkan drilling success ratio dan monitoring reservoir geotermal. Keberadaannya dikontrol oleh fracture akibat stress. Salah satu metode untuk menentukan keberadaan zona permeabel yang dikontrol oleh rekahan atau patahan adalah MEQ microearthquake . Identifikasi dan analisis karakteristik fracture dapat digunakan untuk mengoptimalkan produktivitas. Data gempa mikro tidak hanya memetakan sebaran zona permeabel berdasarkan sebaran hiposenternya, tetapi juga mampu mengkarakterisasi zona fracture berdasarkan analisis mekanisme fokal dan momen tensor. Dari data MEQ lapangan 'X' dengan memanfaatkan waveform lokal tiga komponen telah dilakukan inversi momen tensor. Hasil penelitian menunjukkan bahwa sebaran fracture yang mengontrol permeabilitas memiliki dominasi arah orientasi strike yakni NW-SE dan NE-SW. Hasil analisis momen tensor menunjukkan pada lapangan bagian Utara di elevasi sekitar 1 km bsl ke atas didominasi komponen implosif, berkaitan dengan pergerakan batuan secara konvergen yang dapat berdampak pada potensi penurunan permeabilitas batuan reservoir. Lapangan bagian Utara di elevasi sekitar 1 km bsl ke bawah menunjukkan komponen-komponen eksplosif, berkaitan dengan pergerakan batuan secara divergen yang mengindikasikan distribusi permeabilitas di lapangan Utara secara keseluruhan tergolong baik. Namun tetap ada potensi dan indikasi penurunan permeabilitas karena jika pergerakan konvergen hasil komponen implosif terus terjadi akibat ekstraksi massa fluida dan tidak diimbangi dengan suplai fluida ke reservoir, maka akan berpengaruh pada sifat fisik reservoir, termasuk penurunan permeabilitas. Selain itu, hasil penelitian ini juga menunjukkan bahwa permeabilitas di zona Selatan cukup besar.

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Rock permeability is an important parameter in improving drilling success ratio and monitoring of geothermal reservoir. Its existence is controlled by fracture due to stress. Identification and analysis of fracture characteristics can be used to optimize the productivity. MEQ microearthquake is a method that can be used to determine the existing of permeable zones controlled by fractures or faults. MEQ data not only map the permeable zone distribution based on its hypocenter, but also characterize the fracture zones based on analysis of focal mechanism and moment tensor. Moment tensor inversion has done using MEQ data by utilizing three components of local waveform. The results of this study indicate that the distribution of fractures that control permeability has dominant strike orientation direction ie NW SE and NE SW.

The results of moment tensor analysis show in the northern field at elevation of about 1 km bsl upward is dominated by implosive components, related to convergent rock movement which can impact on potential decrease of permeability of reservoir rock. The northern field at elevation of about 1 km bsl down show explosive components, related to diverging rock movement which indicates the distribution of permeability in the North field as a whole is quite good. However, there are potential and indication of a decrease in permeability because if convergent motion continues to occur due to fluid mass extraction and is not balanced with fluid supply to the reservoir, it will affect the physical properties of the reservoir, including the decrease in permeability. In addition, the results of this study also indicate that permeability in the South zone is considerable. "

"Kami mempelajari sifat atau karakteristik penggunaan fungsi kopling A pada teori skalar tensor terhadap massa-radius bintang neutron. Kami membatasi pengerjaan ini untuk kasus bintang statik dengan simetri bola. Teramati, fungsi kopling yang dipilih menghasilkan fenomena spontaneous scalarization pada kasus bintang neutron dengan persamaan keadaan politropik dan persamaan keadaan NL3 pada bintang neutron saat nilai asymp; -4.5. Namun, skalarisasi tidak terjadi untuk EoS G2 dengan hyperon. Lebih lanjut, pada kasus bintang quark fenomena skalarisasi juga terjadi saat ? asymp; -4.5 baik dengan penggunaan EoS Scalar Coulomb maupun EoS Vector Coulomb.

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We study the characteristic of choosing coupling function A in scalar tensor theory to find mass radius relation from the neutron star. We resrict the case for static with spherical simetric star. We observe, the choosing of our coupling function produce spontaneous scalarization phenomenon in neutron star with polytropic equation of state and NL3 equation of state in neutron star when asymp 4.5. But, scalararization does not occur when using G2 with hyperon equation of state. Furthermore, in quark star case scalarization phenomenon also occur when asymp 4.5 for both Scalar Coulomb and Vector Coulomb equation of state."

"[ABSTRAKPenentuan zona rekahan reservoar di daerah Geotermal sangat penting untuk keperluan penentuan titik pemboran. Penentuan zona rekahan tersebut dapat dilakukan dengan menerapkan metode geofisika, salah satunya adalah metode microearthquake (MEQ). Metode MEQ dapat memberikan informasi yang berkaitan dengan struktur permeabilitas reservoar, pola pergerakan fluida injeksi, dan batas reservoar pada lapangan Geotermal. Terdapat beberapa metode penting yang dilakukan untuk analisis zona rekahan dari data MEQ, yaitu relokasi menggunakan metode double difference, tensor momen dan tomografi. Dalam hal ini penulis berupaya untuk melakukan penelitian pengembangan software terkait penentuan waktu tiba menggunakan spektrogram. Setelah lokasi hiposenter diperoleh, maka langkah berikutnya adalah melakukan analisis tensor momen dan tomografi. Dari berbagai analisis yang dilakukan tersebut, penentuan zona rekahan di daerah Geotermal dapat dilakukan dengan baik. Diharapkan penelitian ini memberikan hasil yang terbaik sehingga metode yang dilakukan tersebut dapat diterapkan dalam penentuan zona rekahan yang lebih tepat.

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ABSTRACTDetermination of the reservoir fracture zone in Geothermal areas are very important for the purposes of determining the drilling point. Determination of the fracture zone can be performed by applying geophysical methods, one of which is a microearthquake (MEQ) method. MEQ method may provide information relating to the structure of the reservoir permeability, patterns of fluid injection movement, and boundary the field of Geothermal reservoir. There are several important methods to analyze fracture zone performed on the data MEQ, relocation using the double difference method, moment tensor and tomography. In this case the author seeks to conduct research related to the development of software such methods can be used to process and analyze the MEQ data. In this case I do research related to software development related to the timing of arrival using the spectrogram. After the location of the hypocenter is obtained, then the next step is to analyze the moment tensor and tomography. From the various analyzes performed, the determination of the fracture zone in the Geothermal area can be done well. It is expected that this study provides the best results so the methods can applied in the determination of a more precise fracture zone.;Determination of the reservoir fracture zone in Geothermal areas are very important for the purposes of determining the drilling point. Determination of the fracture zone can be performed by applying geophysical methods, one of which is a microearthquake (MEQ) method. MEQ method may provide information relating to the structure of the reservoir permeability, patterns of fluid injection movement, and boundary the field of Geothermal reservoir. There are several important methods to analyze fracture zone performed on the data MEQ, relocation using the double difference method, moment tensor and tomography. In this case the author seeks to conduct research related to the development of software such methods can be used to process and analyze the MEQ data. In this case I do research related to software development related to the timing of arrival using the spectrogram. After the location of the hypocenter is obtained, then the next step is to analyze the moment tensor and tomography. From the various analyzes performed, the determination of the fracture zone in the Geothermal area can be done well. It is expected that this study provides the best results so the methods can applied in the determination of a more precise fracture zone.;Determination of the reservoir fracture zone in Geothermal areas are very important for the purposes of determining the drilling point. Determination of the fracture zone can be performed by applying geophysical methods, one of which is a microearthquake (MEQ) method. MEQ method may provide information relating to the structure of the reservoir permeability, patterns of fluid injection movement, and boundary the field of Geothermal reservoir. There are several important methods to analyze fracture zone performed on the data MEQ, relocation using the double difference method, moment tensor and tomography. In this case the author seeks to conduct research related to the development of software such methods can be used to process and analyze the MEQ data. In this case I do research related to software development related to the timing of arrival using the spectrogram. After the location of the hypocenter is obtained, then the next step is to analyze the moment tensor and tomography. From the various analyzes performed, the determination of the fracture zone in the Geothermal area can be done well. It is expected that this study provides the best results so the methods can applied in the determination of a more precise fracture zone., Determination of the reservoir fracture zone in Geothermal areas are very important for the purposes of determining the drilling point. Determination of the fracture zone can be performed by applying geophysical methods, one of which is a microearthquake (MEQ) method. MEQ method may provide information relating to the structure of the reservoir permeability, patterns of fluid injection movement, and boundary the field of Geothermal reservoir. There are several important methods to analyze fracture zone performed on the data MEQ, relocation using the double difference method, moment tensor and tomography. In this case the author seeks to conduct research related to the development of software such methods can be used to process and analyze the MEQ data. In this case I do research related to software development related to the timing of arrival using the spectrogram. After the location of the hypocenter is obtained, then the next step is to analyze the moment tensor and tomography. From the various analyzes performed, the determination of the fracture zone in the Geothermal area can be done well. It is expected that this study provides the best results so the methods can applied in the determination of a more precise fracture zone.]"

"[ABSTRAK Pengaruh dari kopling tensor, isovektor-isoskalar dan pertukaran elektromagnetikterhadap inti berat dan prediksi inti super berat dipelajari melalui teori RelativisticMean Field (RMF). Ditemukan bahwa ketiga kopling tersebut memberikan penga-ruh yang signikan terhadap sifat materi nuklir, energi ikat, spektrum energi partikeltunggal, distribusi kerapatan dan ketebalan kulit inti berat dan prediksi inti superberat. Hasil Penelitian ini menunjukan bahwa 208Pb dan 292120 adalah double ma-gic nuclei. hasil ini konsisten dengan data eksperimen dan prediksi model RMF,sedangkan hasil utama penelitian ini nememukan bahwa 276U adalah double magicnuclei.ABSTRACT We studied the eect of tensor, isovektor-isoskalar and electromagnetic exchange

couplings on heavy and prediction of superheavy nuclei by using RMF model. We

have found that those couplings signicantly in

uence the nuclear matter properties,

binding energy, single particle spectra, nucleon densities and neutron skin of heavy

and superheavy nuclei. Our result show that 208Pb and 292120 are double magic

nuclei. This results are consistent with experimental nding and other RMF model

prediction, while our main result is 276U is also double magic nuclei.;We studied the eect of tensor, isovektor-isoskalar and electromagnetic exchange

couplings on heavy and prediction of superheavy nuclei by using RMF model. We

have found that those couplings signicantly in

uence the nuclear matter properties,

binding energy, single particle spectra, nucleon densities and neutron skin of heavy

and superheavy nuclei. Our result show that 208Pb and 292120 are double magic

nuclei. This results are consistent with experimental nding and other RMF model

prediction, while our main result is 276U is also double magic nuclei.;We studied the eect of tensor, isovektor-isoskalar and electromagnetic exchange

couplings on heavy and prediction of superheavy nuclei by using RMF model. We

have found that those couplings signicantly in

uence the nuclear matter properties,

binding energy, single particle spectra, nucleon densities and neutron skin of heavy

and superheavy nuclei. Our result show that 208Pb and 292120 are double magic

nuclei. This results are consistent with experimental nding and other RMF model

prediction, while our main result is 276U is also double magic nuclei., We studied the eect of tensor, isovektor-isoskalar and electromagnetic exchange

couplings on heavy and prediction of superheavy nuclei by using RMF model. We

have found that those couplings signicantly in

uence the nuclear matter properties,

binding energy, single particle spectra, nucleon densities and neutron skin of heavy

and superheavy nuclei. Our result show that 208Pb and 292120 are double magic

nuclei. This results are consistent with experimental nding and other RMF model

prediction, while our main result is 276U is also double magic nuclei.]"